Q921 re1 lec6 v1

Reservoir Engineering 1 Course (2nd Ed.)

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Darcy Law: Linear Flow Model
Permeability Measurements
Darcy Law: Radial Flow Model
Permeability-Averaging Techniques
Effective Permeabilities
Rock Compressibility
Homogeneous and Heterogeneous Reservoirs
Two-Phase Permeability
Reservoir Characteristics

1. Reservoir Characteristics
A.
B.
C.
D.

Reservoir Fluid Types According To Compressibility
Types of Flow Regimes
Types of Reservoir Geometries
Darcy’s Law Remarks

2. SS Regime for:
A. Linear Flow and Tilted Reservoirs
B. Radial Flow of
a. Incompressible and Slightly Compressible Fluids
b. Compressible Fluids

Types of Fluids in the Reservoir
The isothermal compressibility coefficient is
essentially the controlling factor in identifying the
type of the reservoir fluid.

 In general, reservoir fluids are classified into three
groups:
Incompressible fluids
Slightly compressible fluids
Compressible fluids
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Reservoir Engineering 1 Course: Fundamentals of Reservoir Fluid Flow and SS Regime

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Incompressible Fluids
Incompressible fluids do not exist; this behavior,
however, may be assumed in some cases to simplify
the derivation and the final form of many flow
equations.
An incompressible fluid is defined as the fluid
whose volume (or density) does not change with
pressure, i.e.:

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Slightly Compressible Fluids
These “slightly” compressible fluids exhibit small
changes in volume, or density, with changes in
pressure.
Knowing the volume Vref of a slightly compressible
liquid at a reference (initial) pressure pref, the changes in
the volumetric behavior of this fluid as a function of
pressure p can be mathematically described by:

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Slightly Compressible Fluids (Cont.)
It should be pointed out that crude oil and water
systems fit into this category.
The ex may be represented by a series expansion
as:
Because the exponent x [which represents the term c
(pref−p)] is very small, the ex term can be approximated
by truncating to ex = 1 + x

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Compressible Fluids
These are fluids that experience large changes in
volume as a function of pressure.
All gases are considered compressible fluids.
The truncation of the series expansion is not valid in this
category and the complete expansion is used.

The isothermal compressibility of any compressible
fluid is described by the following expression:

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Schematic Illustrations of
the V and ρ vs. P

Pressure-volume relationship
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Fluid density vs. p for different fluid types


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Flow Regimes
There are basically three types of flow regimes that
must be recognized in order to describe the fluid
flow behavior and reservoir pressure distribution as
a function of time.
There are three flow regimes:
Steady-state flow
Unsteady-state flow
Pseudosteady-state flow

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Steady-State Flow
The flow regime is identified as a steady-state flow
if the pressure at every location in the reservoir
remains constant, i.e., does not change with time.
Mathematically, this condition is expressed as:

In reservoirs, the steady-state flow condition can
only occur when the reservoir is completely
recharged and supported by strong aquifer or
pressure maintenance operations.
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Unsteady-State Flow
The unsteady-state flow
(frequently called transient flow)
 is defined as the fluid flowing condition at which the
rate of change of pressure with respect to time at any
position in the reservoir is not zero or constant.

This definition suggests that the pressure derivative
with respect to time is essentially a function of both
position i and time t, thus

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Pseudosteady-State Flow
When the pressure at different locations in the
reservoir is declining linearly as a function of time,
i.e., at a constant declining rate, the flowing
condition is characterized as the pseudosteadystate flow.
It should be pointed out that the pseudosteadystate flow is commonly referred to as semisteadystate flow and quasisteady-state flow.

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Schematic Comparison of
Flow Regimes
Figure shows
a schematic
comparison of
the pressure
declines as a
function of
time of the
three flow
regimes.

Flow regimes
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Reservoir Geometry
The shape of a reservoir has a significant effect on
its flow behavior.
Most reservoirs have irregular boundaries and a rigorous
mathematical description of geometry is often possible
only with the use of numerical simulators.
For many engineering purposes, however, the actual
flow geometry may be represented by one of the
following flow geometries:
Radial flow
Linear flow
Spherical and hemispherical flow

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Radial Flow
In the absence of severe reservoir
heterogeneities, flow into or away
from a wellbore will follow radial
flow lines from a substantial
distance from the wellbore.
Because fluids move toward the well
from all directions and coverage at
the wellbore, the term radial flow is
given to characterize the flow of
fluid into the wellbore.
Figure shows idealized flow lines
and iso-potential lines for a radial
flow system.
Ideal radial flow
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Linear Flow
Linear flow occurs when
flow paths are parallel
and the fluid flows in a
single direction.
In addition, the cross
sectional area to flow
must be constant.
A common application of
linear flow equations is
the fluid flow into vertical
hydraulic fractures.
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Spherical and Hemispherical Flow
Depending upon the type
of wellbore completion
configuration, it is possible
to have a spherical or
hemispherical flow near the
wellbore.
 A well with a limited
perforated interval could
result in spherical flow in the
vicinity of the perforations.
 A well that only partially
penetrates the pay zone,
could result in hemispherical
flow. The condition could
arise where coning of
bottom water is important.
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Number of
Flowing Fluids in the Reservoir
The mathematical expressions that are used to
predict the volumetric performance and pressure
behavior of the reservoir vary in forms and
complexity depending upon the number of mobile
fluids in the reservoir.
Single-phase flow (oil, water, or gas)
Two-phase flow (oil-water, oil-gas, or gas-water)
Three-phase flow (oil, water, and gas)

The description of fluid flow and subsequent
analysis of pressure data becomes more difficult as
the number of mobile fluids increases.
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Fluid Flow Equations
The fluid flow equations that are used to describe
the flow behavior in a reservoir can take many
forms depending upon the combination of variables
presented previously,
(i.e., types of flow, types of fluids, etc.).

By combining the conservation of mass equation
with the transport equation (Darcy’s equation) and
various equations-of-state, the necessary flow
equations can be developed.

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Darcy’s Law Circumstances
Darcy’s Law applies only when the following
conditions exist:
Laminar (viscous) flow
Steady-state flow
Incompressible fluids
Homogeneous formation

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Deviation from Laminar Flow
in Darcy’s Law
For turbulent flow, which occurs at higher
velocities, the pressure gradient increases at a
greater rate than does the flow rate and a special
modification of Darcy’s equation is needed.
When turbulent flow exists, the application of
Darcy’s equation can result in serious errors.

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Steady-State Flow
The applications of the steady-state flow are:
Linear flow of :
incompressible fluids
slightly compressible fluids
compressible fluids (gases)

Radial flow of :
incompressible fluids
slightly compressible fluids
compressible fluids

Multiphase flow

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SS Regime for Linear Flow of IC Fluids

It is desirable to
express the above
relationship in
customary field units,
or:

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Where q = flow rate,
bbl/day
k = absolute
permeability, md
p = pressure, psia
μ = viscosity, cp
L = distance, ft
A = cross-sectional
area, ft2


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Fluid Flow in Tilted Reservoirs
The difference in the pressure (p1−p2) in Darcy’s
Equation is not the only driving force in a tilted
reservoir.
The gravitational force is the other important
driving force that must be accounted for to
determine the direction and rate of flow.
The fluid gradient force (gravitational force) is always
directed vertically downward while the force that results
from an applied pressure drop may be in any direction.

The force causing flow would then be the vector
sum of these two.
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Fluid Potential
In practice, we obtain this result by introducing a
new parameter, called fluid potential, which has the
same dimensions as pressure, e.g., psi. Its symbol is
Φ.
The fluid potential at any point in the reservoir is
defined as the pressure at that point less the
pressure that would be exerted by a fluid head
extending to an arbitrarily assigned datum level.

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Fluid Potential (Cont.)
Letting Δzi be the vertical distance from a point i in
the reservoir to this datum level.

Where ρ is the density in lb/ft3.
Expressing the fluid density in gm/cc

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Datum in Fluid Potential
The datum is usually selected at the gas-oil contact,
oil-water contact, or at the highest point in
formation.
In using Equations to calculate the fluid potential
Φi at location i, the vertical distance Δzi is assigned
as a positive value when the point i is below the
datum level and as a negative when it is above the
datum level, i.e.:
If point i is above the datum level:
If point i is below the datum level:
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Darcy’s Equation in Tilted Reservoirs
It should be pointed out that
the fluid potential drop (Φ1 − Φ2) is equal to
the pressure drop (p1 − p2)
only when the flow system is horizontal.

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SS Regime for Linear Flow of SC Fluids

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SS Regime for Linear Flow
of C Fluids (Gases)
For a viscous (laminar) gas flow in a homogeneouslinear system, the real-gas equation-of-state can be
applied to calculate the number of gas moles n at
pressure p, temperature T, and volume V:

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SS Regime for Linear Flow
of C Fluids (Gases) (Cont.)
It is essential to notice that those gas properties z
and μg are a very strong function of pressure, but
they have been removed from the integral to
simplify the final form of the gas flow equation.
The above equation is valid for applications when
the pressure < 2000 psi.
The gas properties must be evaluated at the
average pressure p– as defined below.

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SS Regime for Radial Flow of IC Fluids
Because the fluid is incompressible, the flow rate q
must be constant at all radii.
Due to the steady-state flowing condition, the
pressure profile around the wellbore is maintained
constant with time.
Let pwf represent the maintained bottom-hole
flowing pressure at the wellbore radius rw and pe
denote the external pressure at the external or
drainage radius.

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(Cont.)
Darcy’s equation can be used to determine the flow
rate at any radius r:

The flow rate for a crude oil system is customarily
expressed in surface units, i.e., stock-tank barrels
(STB), rather than reservoir units.

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(Cont.)
Where:
Frequently the two
Qo = oil, flow rate,
radii of interest are the
STB/day
wellbore radius rw and
pe = external pressure, psi
the external or drainage
pwf = bottom-hole
flowing pressure, psi
radius re.
k = permeability, md
Then:
μo = oil viscosity, cp
Bo = oil formation volume
factor, bbl/STB
h = thickness, ft
re = external or drainage
radius, ft
rw = wellbore radius, ft
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The External (Drainage) Radius
The external (drainage) radius re is usually
determined from the well spacing by equating the
area of the well spacing with that of a circle, i.e.,

Where A is the well spacing in acres.

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Pressure P at Any Radius R
In practice, neither the external radius nor the
wellbore radius is generally known with precision.
Fortunately, they enter the equation as a logarithm,
so that the error in the equation will be less than
the errors in the radii.

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Average Reservoir Pressure
The external pressure pe cannot be measured
readily, but Pe does not deviate substantially from
initial reservoir pressure if a strong and active
aquifer is present.
The average reservoir pressure pr, which often is
reported in well test results, should be used in
performing material balance calculations and flow rate
prediction.
Craft and Hawkins (1959) showed that the average
pressure is located at about 61% of the drainage radius
re for a steady-state flow condition.

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Oil Flow Rate as a Function of Pr
SS Regime for Radial Flow of IC Fluids

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SS Regime for Radial Flow of SC Fluids
Choosing the bottomhole flow pressure pwf as
the reference pressure
and expressing the flow
rate in STB/day gives:

Where qref is oil flow rate
at a reference pressure
pref.
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Where co = isothermal
compressibility
coefficient, psi−1
Qo = oil flow rate, STB/day


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SS Regime for Radial Flow of C Fluids
The basic differential
form of Darcy’s Law for
a horizontal laminar
flow is valid for
describing the flow of
both gas and liquid
systems. For a radial
gas flow, the Darcy’s
equation takes the
form:

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Where:
 qgr = gas flow rate at
radius r, bbl/day
r = radial distance, ft
h = zone thickness, ft
μg = gas viscosity, cp
p = pressure, psi
0.001127 = conversion
constant from Darcy
units to field units


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(Cont.)
The gas flow rate is usually expressed in scf/day.
Referring to the gas flow rate at standard condition
as Qg, the gas flow rate qgr under pressure and
temperature can be converted to that of standard
condition by applying the real gas equation-of-state
to both conditions, or

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(Cont.)
Integrating from the wellbore conditions (rw and
pwf) to any point in the reservoir (r and p) to give:

Imposing Darcy’s Law conditions on the Equation,
i.e.:
Steady-state flow, which requires that Qg is constant at
all radii
Homogeneous formation, which implies that k and h are
constant
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Real Gas Potential or
Real Gas Pseudopressure

The integral is called the real gas potential or real gas
pseudopressure, and it is usually represented by m (p) or
ψ. Thus

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Real Gas Potential or
Real Gas Pseudopressure Calculation
To calculate the integral,
the values of 2p/μgz are
calculated for several
values of pressure p.
Then (2p/μgz) versus p is
plotted on a Cartesian
scale and the area under
the curve is calculated
either numerically or
graphically, where the
area under the curve
from p = 0 to any
pressure p represents the
value of ψ corresponding
to p.
Real gas pseudopressure data
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Gas Flow Rate in scf/Day
The Equation indicates that a graph of ψ vs. ln r/rw
yields a straight line of slope (QgT/0.703kh) and
intercepts ψw. The flow rate is given exactly by

In the particular case when r = re, then: (Qg [scf/D]

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Gas Flow Rate in Mscf/Day
The gas flow rate is
commonly expressed in
Mscf/day, or

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Where:
ψe = real gas potential
as evaluated from 0 to
pe, psi2/cp
ψw = real gas potential
as evaluated from 0 to
Pwf, psi2/cp
h = thickness, ft
re = drainage radius, ft
rw = wellbore radius, ft
Qg = gas flow rate,
Mscf/day


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Exact Gas Flow Rate in Terms of Pr
The Equation can be expressed in terms of the
average reservoir pressure pr instead of the initial
reservoir pressure pe as:

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Approximation of
the Gas Flow Rate (P2 Method)
The exact gas flow rate as expressed by the
different forms of Darcy’s Law, can be approximated
by removing the term 2/μgz outside the integral as
a constant.
It should be pointed out that the zμg is considered
constant only under a pressure range of < 2000 psi.

The term (μg z) avg is evaluated at an average pressure
p– that is defined by the following expression:
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1. Ahmed, T. (2010). Reservoir engineering
handbook (Gulf Professional Publishing).
Chapter 6

1. Multiple-Phase Flow
2. Pressure Disturbance in Reservoirs
3. USS Flow Regime
A. USS: Mathematical Formulation

4. Diffusivity Equation
A. Solution of Diffusivity Equation
a.

Ei- Function Solution

Q921 re1 lec6 v1

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